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Moscow-Beijing Topology seminar
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[Two combinatorial problems on the triangular grid and their connections with topology] |
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Аннотация: Problems arising in other areas of mathematics can sometimes be reformulated in purely combinatorial terms, and topological perspectives and methods can play an important role in studying such problems. This talk presents two such problems on the triangular grid. Here, the triangular grid refers to the subdivision of an equilateral triangle into k^2 congruent equilateral triangles. The first asks how to assign distinct integers to the vertices of a triangular grid so that the largest difference between the integers assigned to adjacent vertices is minimized. This is called the bandwidth problem, which has natural applications in areas such as coding and data transmission. I will present its exact solution and explain a proof by Hochberg, McDiarmid, and Saks based on Sperner's lemma. The second asks how to color the vertices of a triangular grid with two colors so that the number of monochromatic downward-pointing elementary triangles is as small as possible. This problem arises from an algebraic question concerning the ranks of certain polynomials. I will present some results from joint work with Yaokun Wu and Pengyu Zhu. Some of our arguments already have a topological flavor, and I will discuss the possibility of applying further topological methods. Язык доклада: английский Website: https://us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09 |
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